Fundamentals of Machine Design P37

Instructional Objectives: At the end of this lesson, the students should be able to understand: • Uses and advantages of belt drives • Types of belt drives and their nomenclature • Relat

Module

13 Belt drives

Lesson

1 Introduction to Belt

drives

Instructional Objectives:

At the end of this lesson, the students should be able to understand:

• Uses and advantages of belt drives

• Types of belt drives and their nomenclature

• Relationship between belt tensions

• Some commonly used design parameters

13.1.1 Flexible Machine Elements

Belt drives are called flexible machine elements Flexible machine elements are used for a large number of industrial applications, some of them are as follows

1 Used in conveying systems

Transportation of coal, mineral ores etc over a long distance

2 Used for transmission of power

Mainly used for running of various industrial appliances using prime movers

like electric motors, I.C Engine etc

3 Replacement of rigid type power transmission system

A gear drive may be replaced by a belt transmission system

Flexible machine elements has got an inherent advantage that, it can absorb a good amount of shock and vibration It can take care of some degree of misalignment between the driven and the driver machines and long distance power transmission, in comparison to other transmission systems, is possible For all the above reasons flexible machine elements are widely used in industrial application

Although we have some other flexible drives like rope drive, roller chain drives etc we will only discuss about belt drives

13.1.2 Typical belt drives

Two types of belt drives, an open belt drive, (Fig 13.1.1) and a crossed belt drive (Fig 13.1.2) are shown In both the drives, a belt is wrapped around the pulleys Let us consider the smaller pulley to be the driving pulley This pulley will transmit motion to the belt and the motion of the belt in turn will give a rotation to the larger driven pulley In open belt drive system the rotation of both the pulleys is in the same direction, whereas, for crossed belt drive system, opposite direction of rotation is observed

13.1.3 Nomenclature of Open Belt Drive

dL - Diameter of the larger pulley

dS – Diameter of the smaller pulley

αL- Angle of wrap of the larger pulley

αS – Angle of wrap of the smaller pulley

L

C- Center distance between the two pulleys

Basic Formulae

αL = 180ο + 2β

αS = 180ο - 2β

Where angle β is,

L0 = Length of open belt

This formulae may be verified by simple geometry

13.1.4 Nomenclature of Cross Belt Drive

dL - Diameter of the larger pulley

dS – Diameter of the smaller pulley

αL- Angle of wrap of the larger pulley

αS – Angle of wrap of the smaller pulley

C- Center distance between the two pulleys

C

β

ds

dL

Fig.13.1.1 Open belt drive

1 d L d S

sin

2C

o L S L

1

π

Fig 13.1.2 Cross belt drive

L

β

C

ds

dL

Basic Formulae

α L = α S = 180ο + 2β

Where angle β is,

1 d L d S

sin

2C

Length of cross belt

( ) ( )2

1

π

13.1.5 Belt tensions

The belt drives primarily operate on the friction principle i.e the friction between the belt and the pulley is responsible for transmitting power from one pulley to the other In other words the driving pulley will give a motion to the belt and the motion of the belt will be transmitted to the driven pulley Due to the presence of friction between the pulley and the belt surfaces, tensions on both the sides of the belt are not equal So it is important that one has to identify the higher tension side and the lower tension side, which is shown in Fig 13.1.3

Belt motion

Fig.13.1.3 Belt tensions

Driving pulley Driven pulley

Friction

on pulley

Friction

on belt

T1>T2

When the driving pulley rotates (in this case, anti-clock wise), from the fundamental concept of friction, we know that the belt will oppose the motion of

the pulley Thereby, the friction, f on the belt will be opposite to the motion of the

pulley Friction in the belt acts in the direction, as shown in Fig 13.1.3, and will

impart a motion on the belt in the same direction The friction f acts in the same

direction as T2 Equilibrium of the belt segment suggests that T1 is higher than

T2 Here, we will refer T1 as the tight side andT2 as the slack side, ie, T1 is higher tension side andT2 is lower tension side

Continuing the discussion on belt tension, the figures though they are continuous, are represented as two figures for the purpose of explanation The driven pulley in the initial stages is not rotating The basic nature of friction again suggests that the driven pulley opposes the motion of the belt The directions of friction on the belt and the driven pulley are shown the figure The frictional force

on the driven pulley will create a motion in the direction shown in the figure Equilibrium of the belt segment for driven pulley again suggests that T1 is higher than T2

It is observed that the slack side of the belt is in the upper side and the tight side

of the belt is in the lower side The slack side of the belt, due to self weight, will not be in a straight line but will sag and the angle of contact will increase However, the tight side will not sag to that extent Hence, the net effect will be an increase of the angle of contact or angle of wrap It will be shown later that due to the increase in angle of contact, the power transmission capacity of the drive system will increase On the other hand, if it is other way round, that is, if the slack side is on the lower side and the tight side is on the upper side, for the same reason as above, the angle of wrap will decrease and the power transmission capacity will also decrease Hence, in case of horizontal drive system the tight side is on the lower side and the slack side is always on the upper side

13.1.6 Derivation of relationship between belt tensions

The Fig.13.1.4 shows the free body diagram of a belt segment

dφ

2

dφ

d

2

φ

r dN

dN

μ

CF α

2 2

v centrifugal force(CF) m(rdφ)

r

mv dφ

=

=

=

T1

T2

T

T+dT

Fig.13.1.4

The belt segment subtends an angle dφ at the center Hence, the length of the

belt segment,

dl = r dφ

(13.1.1)

At the impending condition, ie., when the belt is in just in motion with respect to

the pulley, the forces acting on the belt segment are shown in Fig.13.1.4 This

belt segment is subjected to a normal force acting from the pulley on the belt

segment and due to the impending motion the frictional force will be acting in the

direction as shown in the figure

f = μdl

(13.1.2)

where μ is the coefficient of friction between the belt and the pulley

The centrifugal force due to the motion of the belt acting on the belt segment is

denoted as CF and its magnitude is,

CF = [m(rdφ)x v 2 ]/r = mv 2 dφ

(13.1.3)

Where, v is the peripheral velocity of the pulley m is the mass of the belt of unit

length,

m = btρ

(13.1.4)

where, b is the width, t is the thickness and ρ is the density of the belt material

From the equation of equilibrium in the tangential and normal direction,

− + + μ = (13.1.5)

t

F 0

∑ =

2

mv d dN T sin d T dT sin d 0

2 2

⎝ ⎠

(13.1.6)

n

For small angle, dφ,

d

cos 1 and

2

φ

φ

≈

Therefore, simplified form of (13.1.5) is,

dN= dT

μ (13.1.7)

From (13.1.6) and using (13.1.7),

2 dT

μ

or, dT 2

d

T mv = μ φ

−

(13.1.8)

Considering entire angle of wrap,

2

1

T

2 T

dT

d

T mv

α

= μ φ

−

∫ ∫0 (13.1.9)

The final equation for determination of relationship between belt tensions is,

2 1

2 2

e

μα

−

=

−

(13.1.10)

It is important to realize that the pulley, driven or driver, for which the product, μα

of (13.1.10) is the least, should be considered to determine the tension ratio

Here, α should be expressed in radians

13.1.7 Elastic Creep and Initial Tension

Presence of friction between pulley and belt causes differential tension in the

belt This differential tension causes the belt to elongate or contract and create a

relative motion between the belt and the pulley surface This relative motion

between the belt and the pulley surface is created due to the phenomena known

as elastic creep

The belt always has an initial tension when installed over the pulleys This initial

tension is same throughout the belt length when there is no motion During

rotation of the drive, tight side tension is higher than the initial tension and slack

side tension is lower than the initial tension When the belt enters the driving pulley it is elongated and while it leaves the pulley it contracts Hence, the driving pulley receives a larger length of belt than it delivers The average belt velocity

on the driving pulley is slightly lower than the speed of the pulley surface On the other hand, driven pulley receives a shorter belt length than it delivers The average belt velocity on the driven pulley is slightly higher than the speed of the pulley surface

Let us determine the magnitude of the initial tension in the belt

Tight side elongation ∝ (T 1 – T i )

Slack side contraction ∝ (T i – T 2 )

Where, T i is the initial belt tension

Since, belt length remains the same, ie, the elongation is same as the contraction,

i T 1 T 2

T

2

+

= (13.1.11)

It is to be noted that with the increase in initial tension power transmission can be

increased If initial tension is gradually increased then T 1 will also increase and at

the same time T 2 will decrease Thus, if it happens that T 2 is equal to zero, then

T1 = 2Ti and one can achieve maximum power transmission

13.1.8 Velocity ratio of belt drive

Velocity ratio of belt drive is defined as,

L S (

S L

d t N

1 s

N = d +t )

+ −

(13.1.12)

where,

N L and N S are the rotational speeds of the large and the small pulley

respectively, s is the belt slip and t is the belt thickness

13.1.9 Power transmission of belt drive

Power transmission of a belt drive is expressed as,

P = ( T 1 – T2 )v

(13.1.13)

where,

P is the power transmission in Watt and v is the belt velocity in m/s

Sample problem

A pump is driven by an electric motor through a open type flat belt drive

Determine the belt specifications for the following data

Motor pulley diameter(dS) = 300 mm, Pump pulley diameter(dL) = 600 mm

Coefficient of friction (μS) for motor pulley = 0.25

Coefficient of friction (μL) for pump pulley = 0.20

Center distance between the pulleys=1000 mm; Rotational speed of the

motor=1440 rpm;

Power transmission = 20kW; density of belt material (ρ)= 1000 kg/m3 ; allowable

stress for the belt material (σ) = 2 MPa; thickness of the belt = 5mm

Solution

0 L

0 S

2

2

D e t e r m i n a t i o n o f a n g l e o f w r a p

s i n ( ) 8 6 3

2 C

1 8 0 2 1 9 7 2 5 3 4 4 r a d

1 8 0 2 1 6 2 7 5 2 8 4 r a d

L e n g t h o f o p e n b e l t

1

1

= 6 0 0 3 0 0 2 0 0 0 6 0 0 3 0 0 = 3 4 3 6 m m

β

π

π

3 3

2

s s

L L

0.688 1

2

300 1440

60 1000

m bt 10 0.005kg / m

10 10

mv 2.56 b N

Now,

0.25 2.84 0.71

0.20 3.44 0.688

l arg er pulley governs the design

T 2.56b

e 1.99

T 2.56b

π

ρ

μ α

μ α

× ×

×

= ×

−

1 2

1 2

1

( 1 )

power equation

P ( T T ) v

putting data,

( T T ) 884.17 N ( 2 )

again,T 2 b 5N

10bN ( from permissible stress ) ( 3 )

From

∴

= × ×

=

(1),( 2 ) and ( 3 ), solving for b,

b 240 mm

Hence,the required belt dim ensions are,

Length 3436 mm; breadth 240mm and thickness 5mm

≈

Questions and answers

Q1 What are the advantages of a belt drive?

A1 The advantages of a belt drive are that, it can absorb a good amount of shock and vibration It can take care of some degree of misalignment between the driven and the driver machines and long distance power transmission, in comparison to other transmission systems, is possible Q2 Why the slack side of the belt of a horizontal belt drive is preferable to place

on the top side?

A2 The slack side of the belt is preferably placed on the top side because, the slack side of the belt, due to its self weight, will sag For this reason the angle of contact between the belt and the pulleys will increase However, the tight side will not sag to that extent Hence, the net effect will be an increase in the angle of contact or angle of wrap Thus, due to the increase

in angle of contact, the power transmission capacity of the drive system will increase

Q3 Which one should be the governing pulley to calculate tension ratio?

A3 The pulley, driven or driver, for which the product, μα of equation for belt

tension is the least, should be considered to determine the tension ratio

References

1 V.Maleev and James B Hartman , Machine Design, CBS Publishers And Distributors.3rd Edition 1983

2 J.E Shigley and C.R Mischke , Mechanical Engineering Design , McGraw Hill Publication, 5th Edition 1989

3 M.F Spotts, Design of Machine Elements, Prentice Hall India Pvt Limited,

6th Edition, 1991

4 Khurmi, R.S and Gupta J.K., Text book on Machine Design, Eurasia

Publishing House, New Delhi

5 Sharma, C.S and Purohit Kamalesh, Design of Machine Elements,

Prentice Hall of India, New Delhi, 2003